Question

find the coefficient of volume expansion for an ideal gas at constant pressure​

Answer 1

Answer:

Coefficient of volume expansion:

$\implies \rm \dfrac{\Delta V}{V} = \gamma \Delta \theta$

$\implies \rm \gamma = \dfrac{\Delta V}{ V\Delta \theta}$

$\implies \purple{ \rm \gamma = \dfrac{1}{v} . \dfrac{dv}{dT} \: \: ....(i)}$

By ideal gas equation:

$\implies PV = nRT$

Taking derivative on both sides w.r.t to T:

$\implies \dfrac{d}{dT} (PV) = \dfrac{d}{dT}(nRT)$

$\implies P.\dfrac{dV}{dT} = nR$

$\implies \red{ \rm \dfrac{dV}{dT} = \dfrac{nR}{P} \: \: \: ...(ii)}$

Substituting the value of dV/dT from equation (ii) to equation (i) :

$\implies \gamma = \dfrac{1}{v} . \dfrac{nR}{P}$

$\implies \gamma = \dfrac{nR}{PV}$

$\implies \gamma = \dfrac{nR}{nRT}$

$\implies \underline{ \boxed{ \orange{ \bf \gamma = \dfrac{1}{T} }}}$